in terms of a slowly varying magnetic field \(H\left( x\right) =H_{0}+\triangle H\left( x\right) ,\)  non-equilibrium magnon chemical potential or
magnon accumulation \(\mu _{m}\left( x\right) \)  and temperature \(T_{m}\left(x\right) =T_{0}+\triangle T_{m}\left( x\right) \) . In magnetic insulators,
this assumption can be justified by efficient magnon-conserving magnon-phonon
and magnon-magnon scattering, while magnon-non-conserving damping processes
are weak [1]}. The magnon number density is then reduced to
\(-m\left( x\right) =\int f_{BE}\left( x,\varepsilon \right) \rho _{m}\left(\varepsilon \right) \frac{d\varepsilon }{2\pi }, \) 
